let X be set ; :: thesis:
let G be Graph; :: thesis:
let v1, v2 be Vertex of G; :: thesis:
let v9 be Vertex of (AddNewEdge (v1,v2)); :: thesis:
assume that
A1: v9 = v2 and
A2: v1 <> v2 ; :: thesis:
set G9 = AddNewEdge (v1,v2);
set E = the carrier' of G;
set S = the Source of G;
set E9 = the carrier' of (AddNewEdge (v1,v2));
set S9 = the Source of (AddNewEdge (v1,v2));
A3: the carrier' of (AddNewEdge (v1,v2)) = the carrier' of G \/ { the carrier' of G} by Def7;

let x be object ; :: thesis:

hereby :: thesis:

assume A4: x in Edges_Out (v9,X) ; :: thesis:
then A5: x in X by Def2;
A6: the Source of (AddNewEdge (v1,v2)) . x = v9 by A4, Def2;
the Source of (AddNewEdge (v1,v2)) . the carrier' of G = v1 by Th34;
then not x in { the carrier' of G} by A1, A2, A6, TARSKI:def 1;
then A7: x in the carrier' of G by A3, A4, XBOOLE_0:def 3;
then the Source of G . x = v2 by A1, A6, Th35;
hence x in Edges_Out (v2,X) by A5, A7, Def2; :: thesis:

end;

assume A8: x in Edges_Out (v2,X) ; :: thesis:
then the Source of G . x = v2 by Def2;
then A9: the Source of (AddNewEdge (v1,v2)) . x = v9 by A1, A8, Th35;
( x in X & x in the carrier' of (AddNewEdge (v1,v2)) ) by A3, A8, Def2, XBOOLE_0:def 3;
hence x in Edges_Out (v9,X) by A9, Def2; :: thesis:

end;

hence Edges_Out (v9,X) = Edges_Out (v2,X) by TARSKI:2; :: thesis: